FFT and multigrid accelerated integral equation solvers for multi-scale electromagnetic analysis in complex backgrounds

textNovel integral-equation methods for efficiently solving electromagnetic problems that involve more than a single length scale of interest in complex backgrounds are presented. Such multi-scale electromagnetic problems arise because of the interplay of two distinct factors: the structure under study and the background medium. Both can contain material properties (wavelengths/skin depths) and geometrical features at different length scales, which gives rise to four types of multi-scale problems: (1) twoscale, (2) multi-scale structure, (3) multi-scale background, and (4) multi-scale-squared problems, where a single-scale structure resides in a different single-scale background, a multi-scale structure resides in a single-scale background, a single-scale structure resides in a multi-scale background, and a multi-scale structure resides in a multi-scale background, respectively. Electromagnetic problems can be further categorized in terms of the relative values of the length scales that characterize the structure and the background medium as (a) high-frequency, (b) low-frequency, and (c) mixed-frequency problems, where the wavelengths/skin depths in the background medium, the structure’s geometrical features or internal wavelengths/skin depths, and a combination of these three factors dictate the field variations on/in the structure, respectively. This dissertation presents several problems arising from geophysical exploration and microwave chemistry that demonstrate the different types of multi-scale problems encountered in electromagnetic analysis and the computational challenges they pose. It also presents novel frequency-domain integral-equation methods with proper Green function kernels for solving these multi-scale problems. These methods avoid meshing the background medium and finding fields in an extended computational domain outside the structure, thereby resolving important complications encountered in type 3 and 4 multi-scale problems that limit alternative methods. Nevertheless, they have been of limited practical use because of their high computational costs and because most of the existing ‘fast integral-equation algorithms’ are not applicable to complex Green function kernels. This dissertation introduces novel FFT, multigrid, and FFT-truncated multigrid algorithms that reduce the computational costs of frequency-domain integral-equation methods for complex backgrounds and enable the solution of unprecedented type 3 and 4 multi-scale problems. The proposed algorithms are formulated in detail, their computational costs are analyzed theoretically, and their features are demonstrated by solving benchmark and challenging multi-scale problems.Electrical and Computer Engineerin

Two-dimensional modeling and inversion of the controlled-source electromagnetic and magnetotelluric methods using finite elements and full-space PDE-constrained optimization strategies

[eng] The controlled-source electromagnetics (CSEM) and magnetotellurics (MT) methods are common geophysical tools for imaging the Earth's electrical interior. To appreciate measured data, both methods require forward and inverse modeling of the subsurface with the ultimate goal of finding a feasible model for which the simulated data reasonably fits the observations. The goodness of this fit depends on the error in the measured data, on the numerical error and on the degree of approximation inferred by numerical modeling. Therefore, active research focuses on new methods for modeling and inversion to improve accuracy and reliability for increasingly complex scenarios. In a first step, physical factors such as anisotropy, topography and realistic sources must be taking into account. Second, numerical methods need to be assessed in terms of solution accuracy, time efficiency and memory demand. The finite elements (FE) methods offer much flexibility in model geometry and contain quality control mechanisms for the solution, as shape function order and adaptive mesh refinement. Most emerging modeling programs are based on FE, however, inversion programs are generally based on finite differences (FD) or integral equation (IE) methods. On the other hand, inverse modeling is usually based on gradient methods and formulated in the reduced-space, where the electrical conductivity is the only optimization variable. Originally, the inverse problem is stated for the EM fields and the conductivity parameter (in the full-space), and constrained by governing partial differential equations (PDEs). The reduced-space strategy eliminates the field variables by applying equality constraints and solves then, the unconstrained problem. A common drawback of this is the repeated costly computation of the forward solution. Solving the PDE-constrained optimization problem directly, in the full-space, has the advantage that it is only necessary to exactly solve the PDEs at the end of the optimization, but it comes at the cost of a larger number of variables. This thesis develops a robust and versatile adaptive unstructured mesh FE program to model the total field for two-dimensional (2-D) anisotropic CSEM and MT data, allowing for arbitrarily oriented, three-dimensional (3-D) sources, for which a two-and-a-half-dimensional (2.5-D) approximation is employed. The formulations of the problems in a FE framework are derived for isotropic and anisotropic subsurface conductivity structures. The accuracy of the solution is controlled and improved by a goal-oriented adaptive mesh refinement algorithm. Exhaustive numerical experiments validate the adaptive FE program for both CSEM and MT methods and on land and marine environments. The influence of the model dimensions, mesh design and order of shape functions on the solution accuracy is studied and notably, an outperformance of quadratic shape functions is found (compared to linear and cubic). Several examples demonstrate the effect of complex scenarios on EM data. In particular, we study the distortion caused by: the bathymetry, the orientation and geometry of the sources and the anisotropy, considering vertical and dipping cases. All examples showcase the importance of adequate consideration of these very common physical features of real world data. Further, a formulation for the 2.5-D CSEM inversion as a PDE-constrained optimization in full-space is derived within a FE framework following two strategies: discretize-optimize and optimize-discretize. The discretize-optimize formulation is implemented using a general purpose optimization algorithm. Two examples, a canonical reservoir model and a more realistic marine model with topography, demonstrate the performance of this inversion scheme, recovering in both cases the model’s main structures within an acceptable data misfit. Finally, the optimize-discretize formulation is derived in a FE framework, as a first step towards a development of an inversion scheme using adaptive FE meshes.[cat] El mètode de font electromagnètica controlada (CSEM) i el mètode magnetotel.lúric (MT) són tècniques geofísiques usades habitualment per obtenir una imatge de les propietats elèctriques del subsòl terrestre i s'utilitzen independentment, conjuntament i en combinació amb altres tècniques geofísiques. Per poder interpretar les dades, ambdós mètodes necessiten la modelització directa i inversa de la conductivitat elèctrica del subsòl amb l'objectiu final d'obtenir un model coherent per al qual les dades simulades s'ajustin de forma raonable a les observacions. Naturalment, la qualitat d'aquest ajust no només depèn de l'error en les dades mesurades i de l'error numèric, sinó també del grau en l'aproximació física inferit per la modelització numèrica. D'aquesta manera, les recerques actuals se centren a investigar noves metodologies per a la modelització i inversió, per tal d'obtenir models acurats i fiables de les estructures de la Terra en escenaris cada cop més complexos. Un primer pas és millorar les aproximacions en la modelització tenint en compte factors físics com ara l'anisotropia, la topografia o fonts més realistes. En segon lloc, per tal d'acomodar aquests factors en un programa de modelització i inversió i per poder tractar els conjunts de dades típicament llargs, els mètodes numèrics han de ser avaluats en termes de la precisió de la solució, l'eficiència en temps i la demanda en memòria. Els mètodes de modelització en elements finits (FE) són coneguts per oferir una major flexibilitat en la modelització de la geometria i contenen mecanismes de control de la solució, com ara l'ordre de les funcions forma i la tècnica de refinament adaptatiu de la malla. La majoria de programes de modelització emergents estan basats en els FE, i mostren avantatges significatius, però gairebé tots els programes de modelització inversa, encara avui dia, estan basats en el mètode de les diferències finites (FD) o en el mètode de l'equació integral (IE). A més a més, la modelització inversa desenvolupada per a dades electromagnètiques (EM) es basa generalment en mètodes del gradient i es formula en un espai reduït, on les úniques variables d'optimització són els paràmetres del model, és a dir, la conductivitat elèctrica del subsòl. Originalment, el problema invers es formula per als camps EM i per al paràmetre conductivitat, i està constret per les equacions diferencials en derivades parcials (PDEs) que governen les variables camps EM. L'estratègia d'espai reduït elimina les variables camps aplicant lligams d'igualtat i soluciona, doncs, el problema no constret en l'espai reduït dels paràmetres del model. Un desavantatge general d'aquests mètodes és la costosa repetició del càlcul de la solució del problema directe i de la matriu jacobiana de sensibilitats (per mètodes basats en Newton). D'altra banda, també és possible de solucionar el problema invers en l'espai complet de les variables camps EM i del paràmetre conductivitat. Solucionar-hi el problema d'optimització constret per les PDEs té l'avantatge que només és necessari de solucionar exactament el problema directe al final del procés d'optimització, però això comporta el cost addicional de tenir moltes més variables d'optimització i de la presència de lligams d'igualtat. També, en particular, en el marc dels FE, el problema d'optimització constret per les PDEs té l'avantatge afegit d'incloure tècniques sofisticades pròpies dels FE en el procés d'inversió, com ara el refinament adaptatiu de la malla. Aquesta tesi desenvolupa un programa robust i versàtil amb FE i malles irregulars adaptatives per modelar numèricament el camp total de dades CSEM i MT bidimensionals (2D) i anisòtropes, que permet l'ús de fonts tridimensionals (3D) orientades arbitràriament. Per tal de representar fonts CSEM 3D en un model físic 2D, s'utilitza una aproximació dos i mig dimensional (2.5D). Les formulacions FE es deriven per a ambdós mètodes, per a estructures de conductivitat del subsòl isòtropes i anisòtropes. Encara que el cas anisòtrop no és general, inclou anisotropia vertical i de cabussament. La precisió en la solució es controla i millora amb un algoritme de refinament adaptatiu de la malla utilitzant mètodes d'estimació de l'error a posteriori. Una sèrie exhaustiva d'experiments numèrics valida el programa de FE adaptatius per ambdós mètodes, CSEM i MT, i en escenaris terrestres i marins. S'estudia la influència de les dimensions del model, del disseny de la malla i de l'ordre de les funcions forma en l'exactitud de la solució i es troba un comportament notablement superior de les funcions forma quadràtiques comparades amb les lineals o cúbiques. Diferents exemples mostren l'efecte d'escenaris complexos sobre les dades EM, en particular, un model amb batimetria, un model terrestre i un de marí amb fonts orientades i de dimensió finita, un medi amb anisotropia vertical amb un reservori encastat i un altre amb un reservori encastat en una estructura anticlinal. Aquests exemples demostren la importància de considerar adequadament (en termes de modelització directa) característiques físiques com la topografia, l'orientació i geometria de la font i l'anisotropia del medi, que sovint es troben en mesures reals. Juntament amb això, es deriva una formulació per al problema invers 2.5D CSEM com una optimització constreta per les PDEs en l'espai complet i en un marc de FE, seguint dues estratègies diferents: discretització-optimització i optimització-discretització. L'estratègia de discretització-optimització considera que el problema invers es troba en forma discretitzada i deriva les condicions d'optimitat de la Lagrangiana i el pas de Newton. Contràriament, l'aproximació optimització-discretització deriva primer les condicions d'optimitat i el pas de Newton o una aproximació d'aquest, i després discretitza les equacions resultants. La implementació de la formulació discretització-optimització es mostra en dos exemples, un model canònic de reservori i un model marí més realista amb topografia, utilitzant un programa d'optimització de propòsit general, que és una implementació d'un algoritme de programació quadràtica seqüencial (SQP). Encara que no s'utilitza una regularització explícita, l'ús de diferents malles per al paràmetre del model i per a les variables camps, permet recuperar les principals estructures del model i obtenir un ajust de les dades acceptable. Cal dir, però, que l'eficiència en temps i memòria del programa hauria de millorar-se. Finalment, el problema invers 2.5D CSEM es formula com un problema d'optimització constret per les PDEs en l'espai complet i en un marc de FE utilitzant una estratègia d'optimització-discretització i com un primer pas per al desenvolupament d'un esquema d'inversió que utilitzi malles adaptatives de FE

Eletrodo de aterramento HVDC do Rio Madeira - Bipolo 1 : modelagem geoelétrica da crosta terrestre para projeto do eletrodo

Orientador: Sueli Yoshinaga PereiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de GeociênciasResumo: Um sistema de transmissão HVDC é composto por duas Subestações Conversoras, interligadas pela linha HVDC, cada uma com um eletrodo de aterramento separado do seu pátio CC, geralmente localizado de 15 km a 150 km de distância e conectado por meio da linha do eletrodo. Os eletrodos HVDC proporcionam redução de custos e agregam confiabilidade ao sistema de transmissão de energia. Os eletrodos geralmente dissipam na terra a corrente de desequilíbrio do bipolo, entre 20 A a 40 A. No caso de perda de um polo da linha HVDC, a energia pode ser transmitida pelo polo remanescente com retorno pela terra, utilizando os eletrodos de aterramento para a injeção de correntes que chegar a quase 4 kA, o que pode resultar em interferências em uma área ampla, dependendo da estrutura geológica. A seleção dos locais de construção dos eletrodos deve ser realizada dentro de um raio de algumas dezenas de quilômetros ao redor das subestações, nas duas extremidades da linha HVDC. O melhor local em cada extremidade é aquele que apresenta a estrutura geoelétrica com resistividades mais baixas, desde a superfície do solo até pelo menos o meio da crosta. Esta tese apresenta o desenvolvimento do modelo geoelétrico 1D para o eletrodo sul do sistema HVDC do Rio Madeira, bipolo 1, localizado em Araraquara, na Bacia Sedimentar do Paraná, sul do Brasil. O eletrodo é constituído por um anel aproximadamente retangular de poços (cerca de 820 m x 560 m), cada um revestido por tubos de aço com profundidades variáveis, entre 20 m e 40 m de profundidade. O modelo geoelétrico deve ser representativo da média do solo raso, até a profundidade dos poços, combinada com um modelo profundo. A modelagem do solo raso foi desenvolvida a partir de uma campanha de sondagens Schlumberger e da perfilagem por indução de poços de monitoramento perfurados no local. O modelo profundo foi construído a partir de uma campanha magnetotelúrica (MT). Os modelos geoelétricos são aprimorados ao longo do projeto, à medida que mais dados geofísicos e geotécnicos são levantados. O modelo de projeto tem um ajuste final após o comissionamento do eletrodo, pois o desempenho elétrico medido permite um ajuste complementar do desvio estático da curva de resistividades aparentes MT. Uma medição independente do potencial tubo-solo foi feita no gasoduto Bolívia-Brasil, a 26 km do eletrodo, sendo o valor medido comparado com o potencial calculado a partir da simulação do eletrodo com o modelo geoelétrico final, com ambos os valores apresentando boa compatibilidadeAbstract: A HVDC transmission system comprises two Converter Substations, interconnected by the HVDC line, each one requiring a separate grounding electrode for its DC switchyard, which usually is located from 15 km to 150 km away and connected by means of the electrode line. HVDC electrodes allow for cost reduction and add reliability to the energy transmission system. The electrodes usually dissipate into the ground the unbalance current of the bipole, about 20 A to 40 A. In case of the loss of one pole of the HVDC line, the energy can be transmitted by the remaining pole with ground return, using grounding electrodes for the injection into the ground currents that may reach almost 4 kA, which may produce interferences within a wide area, depending on the tectonic setting. The electrodes Site Selection shall be carried up within a radius of some tens of kilometers around the substations at the two ends of the HVDC line. The best site at each end is the one with the geoelectric structure that presents lower resistivities, from soil surface down to at least mid-crust. This thesis presents the development of the 1D geoelectric model for the South electrode of Rio Madeira HVDC system, bipole 1, located at Araraquara, in the Paraná Sedimentary Basin, South of Brazil. The electrode is constituted by an approximately rectangular ring of wells (about 820 m x 560 m), each one lined with steel pipes with varying depths, between 20 m to 40 m deep. The geoelectric model shall represent the average of the shallow ground, down to the depth of the wells, combined with a deep model, down to the mid-crust. The modeling of the shallow ground was developed from a Schlumberger survey and from the induction profiling of monitoring wells drilled in the site. The deep model was built from a magnetotelluric (MT) survey. The models are improved along the project, as more geophysical and geotechnical data are surveyed. The design model has a final adjustment after the electrode commissioning, because the measured electrical performance allows for a complementary adjustment of the MT static deviation. An independent measurement of pipe-to-ground potential was done at the Bolivia-Brazil pipeline, 26 km away from the electrode, which was compared with the potential calculated from the electrode simulation using the final geoelectric model, with both values presenting good compatibilityDoutoradoGeologia e Recursos NaturaisDoutor em Geociência

Advanced Geoscience Remote Sensing

Nowadays, advanced remote sensing technology plays tremendous roles to build a quantitative and comprehensive understanding of how the Earth system operates. The advanced remote sensing technology is also used widely to monitor and survey the natural disasters and man-made pollution. Besides, telecommunication is considered as precise advanced remote sensing technology tool. Indeed precise usages of remote sensing and telecommunication without a comprehensive understanding of mathematics and physics. This book has three parts (i) microwave remote sensing applications, (ii) nuclear, geophysics and telecommunication; and (iii) environment remote sensing investigations

Modeling and inversion of airborne full tensor magnetic gradiometry data in the Thuringian basin and forest

The recent development of airborne full tensor magnetic gradiometer (FTMG) systems, based on superconducting quantum interference devices (SQUID), allows to obtain the full magnetic gradient tensor of the Earth's magnetic field of large areas (10x10 km). This system allows acquiring all components of the magnetic gradient tensor. This tensor exhibits some advantages over conventional airborne magnetic field data, e.g. a higher spatial resolution and additional directional sensitivity. In this work a FTMG system was applied in the framework of the multidisciplinary INFLUINS project (Integrated fluid dynamics in sedimentary basins) in order investigate different areas in the Thuringian Basin and the neighboring highlands. Main goal was to map magnetic lineaments along major fault zones and to demonstrate the advantages of airborne FTMG. Full tensor data sets have been acquired with very low system noise of only 60 (pT/m). Two different case studies are presented: In the first case study a strong magnetic anomaly in the center of the Thuringian Forest, caused by the magmatic intrusion of the Höhenberger dolerite is analyzed, which exhibits indications of a significant remanent magnetization. Multiple magnetization vector inversions were performed using either the full magnetic gradient tensor or only the total field anomaly data. The inversion results are evaluated using magnetization directions acquired by paleomagnetic sampling and available geological information. In the second case study, a small magnetic anomaly was investigated. It was discovered while mapping magnetic anomalies along the Eichenberg-Gotha-Saalfeld fault zone, which is one of the major fault zones in the Thuringian Basin. The detected lineament is interpreted using the components of the magnetic gradient tensor, additional ground based geo-electrical data and available geological information. The inversion of the magnetic gradients revealed a steeply dipping zone of mostly induced magnetization